loos/SRGGW
2
0
Fork 0
Code Issues Pull Requests Projects Releases Wiki Activity

modifs in Sec II

Browse Source
This commit is contained in:
Pierre-Francois Loos 2023-02-14 17:30:01 -05:00
parent ec364f9451
commit e721f107b7
1 changed files with 13 additions and 13 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

26
Manuscript/SRGGW.tex
View File

@ -80,7 +80,7 @@ The family of Green's function methods based on the $GW$ approximation has gaine
Despite this, self-consistent versions still pose challenges in terms of convergence. Despite this, self-consistent versions still pose challenges in terms of convergence.
A recent study \href{https://doi.org/10.1063/5.0089317}{[J. Chem. Phys. 156, 231101 (2022)]} has linked these convergence issues to the intruder-state problem. A recent study \href{https://doi.org/10.1063/5.0089317}{[J. Chem. Phys. 156, 231101 (2022)]} has linked these convergence issues to the intruder-state problem.
In this work, a perturbative analysis of the similarity renormalization group (SRG) approach is performed on Green's function methods. In this work, a perturbative analysis of the similarity renormalization group (SRG) approach is performed on Green's function methods.
The SRG formalism enables us to derive, from first principles, the expression of a new, naturally Hermitian form of the static self-energy that can be employed in quasiparticle self-consistent $GW$ (qs$GW$) calculations. The SRG formalism enables us to derive, from first principles, the expression of a new, naturally hermitian form of the static self-energy that can be employed in quasiparticle self-consistent $GW$ (qs$GW$) calculations.
The resulting SRG-based regularized self-energy significantly accelerates the convergence of qs$GW$ calculations and slightly improves the overall accuracy. The resulting SRG-based regularized self-energy significantly accelerates the convergence of qs$GW$ calculations and slightly improves the overall accuracy.
%\bigskip %\bigskip
%\begin{center} %\begin{center}
@ -174,7 +174,7 @@ The central equation of many-body perturbation theory based on Hedin's equations
\end{equation} \end{equation}
where $\bF$ is the Fock matrix in the orbital basis \cite{SzaboBook} and $\bSig(\omega)$ is (the correlation part of) the $GW$ self-energy. where $\bF$ is the Fock matrix in the orbital basis \cite{SzaboBook} and $\bSig(\omega)$ is (the correlation part of) the $GW$ self-energy.
Both are $K \times K$ matrices with $K$ the number of one-electron orbitals. Both are $K \times K$ matrices with $K$ the number of one-electron orbitals.
Throughout the manuscript, the indices $p,q,r,s$ are general orbitals while $i,j,k,l$ and $a,b,c,d$ refers to occupied and virtual orbitals, respectively. Throughout the manuscript, the indices $p,q,r,s$ are general orbitals while $i,j,k,l$ and $a,b,c,d$ refer to occupied and virtual orbitals, respectively.
The indices $\mu$ and $\nu$ are composite indices, that is, $\nu=(ia)$, referring to neutral (single) excitations. The indices $\mu$ and $\nu$ are composite indices, that is, $\nu=(ia)$, referring to neutral (single) excitations.
The self-energy can be physically understood as a correction to the Hartree-Fock (HF) problem (represented by $\bF$) accounting for dynamical screening effects. The self-energy can be physically understood as a correction to the Hartree-Fock (HF) problem (represented by $\bF$) accounting for dynamical screening effects.
@ -212,10 +212,10 @@ and
are bare two-electron integrals in the spin-orbital basis. are bare two-electron integrals in the spin-orbital basis.
The diagonal matrix $\boldsymbol{\Omega}$ contains the positive eigenvalues of the RPA problen defined in Eq.~\eqref{eq:full_dRPA} and its elements $\Omega_\nu$ appear in Eq.~\eqref{eq:GW_selfenergy}. The diagonal matrix $\boldsymbol{\Omega}$ contains the positive eigenvalues of the RPA problen defined in Eq.~\eqref{eq:full_dRPA} and its elements $\Omega_\nu$ appear in Eq.~\eqref{eq:GW_selfenergy}.
In the Tamm-Dancoff approximation (TDA), one sets $\bB = \bO$ in Eq.~\eqref{eq:full_dRPA} which reduces to a Hermitian eigenvalue problem of the form $\bA \bX = \bX \bOm$ (hence $\bY=0$). In the Tamm-Dancoff approximation (TDA), one sets $\bB = \bO$ in Eq.~\eqref{eq:full_dRPA} which reduces to a hermitian eigenvalue problem of the form $\bA \bX = \bX \bOm$ (hence $\bY=0$).
As mentioned above, because of the frequency dependence of the self-energy, solving exactly the quasiparticle equation \eqref{eq:quasipart_eq} is a rather complicated task. As mentioned above, because of the frequency dependence of the self-energy, solving exactly the quasiparticle equation \eqref{eq:quasipart_eq} is a rather complicated task.
Hence, several approximate schemes have been developed to bypass self-consistency. Hence, several approximate schemes have been developed to bypass full self-consistency.
The most popular strategy is the one-shot (perturbative) $G_0W_0$ scheme, where the self-consistency is completely abandoned, and the off-diagonal elements of Eq.~\eqref{eq:quasipart_eq} are neglected. The most popular strategy is the one-shot (perturbative) $G_0W_0$ scheme, where the self-consistency is completely abandoned, and the off-diagonal elements of Eq.~\eqref{eq:quasipart_eq} are neglected.
Assuming a HF starting point, this results in $K$ quasiparticle equations that read Assuming a HF starting point, this results in $K$ quasiparticle equations that read
\begin{equation} \begin{equation}
@ -231,7 +231,7 @@ These solutions can be characterized by their spectral weight given by the renor
\end{equation} \end{equation}
The solution with the largest weight is referred to as the quasiparticle while the others are known as satellites (or shake-up transitions). The solution with the largest weight is referred to as the quasiparticle while the others are known as satellites (or shake-up transitions).
However, in some cases, Eq.~\eqref{eq:G0W0} can have two (or more) solutions with similar weights, hence the quasiparticle is not well-defined. However, in some cases, Eq.~\eqref{eq:G0W0} can have two (or more) solutions with similar weights, hence the quasiparticle is not well-defined.
These additional solutions with large weights are the previously mentioned intruder states. \cite{Monino_2022} %These additional solutions with large weights are the previously mentioned intruder states.
One obvious drawback of the one-shot scheme mentioned above is its starting point dependence. One obvious drawback of the one-shot scheme mentioned above is its starting point dependence.
Indeed, in Eq.~\eqref{eq:G0W0} we choose to rely on HF orbital energies but this is arbitrary and one could have chosen Kohn-Sham energies (and orbitals) instead. Indeed, in Eq.~\eqref{eq:G0W0} we choose to rely on HF orbital energies but this is arbitrary and one could have chosen Kohn-Sham energies (and orbitals) instead.
@ -242,9 +242,9 @@ However, if one of the quasiparticle equations does not have a well-defined quas
Even at convergence, the starting point dependence is not totally removed as the quasiparticle energies still depend on the initial set of orbitals. \cite{Marom_2012} Even at convergence, the starting point dependence is not totally removed as the quasiparticle energies still depend on the initial set of orbitals. \cite{Marom_2012}
In order to update both the orbitals and their corresponding energies, one must consider the off-diagonal elements in $\bSig(\omega)$. In order to update both the orbitals and their corresponding energies, one must consider the off-diagonal elements in $\bSig(\omega)$.
To avoid solving the non-Hermitian and dynamic quasiparticle equation defined in Eq.~\eqref{eq:quasipart_eq}, one can resort to the qs$GW$ scheme in which $\bSig(\omega)$ is replaced by a static approximation $\bSig^{\qsGW}$. To avoid solving the non-hermitian and dynamic quasiparticle equation defined in Eq.~\eqref{eq:quasipart_eq}, one can resort to the qs$GW$ scheme in which $\bSig(\omega)$ is replaced by a static approximation $\bSig^{\qsGW}$.
Then, the qs$GW$ equations are solved via a standard self-consistent field procedure similar to the HF algorithm where $\bF$ is replaced by $\bF + \bSig^{\qsGW}$. Then, the qs$GW$ equations are solved via a standard self-consistent field procedure similar to the HF algorithm where $\bF$ is replaced by $\bF + \bSig^{\qsGW}$.
Various choices for $\bSig^{\qsGW}$ are possible but the most popular is the following Hermitian approximation Various choices for $\bSig^{\qsGW}$ are possible but the most popular is the following hermitian approximation
\begin{equation} \begin{equation}
\label{eq:sym_qsgw} \label{eq:sym_qsgw}
\Sigma_{pq}^{\qsGW} = \frac{1}{2}\Re[\Sigma_{pq}(\epsilon_p) + \Sigma_{pq}(\epsilon_q) ]. \Sigma_{pq}^{\qsGW} = \frac{1}{2}\Re[\Sigma_{pq}(\epsilon_p) + \Sigma_{pq}(\epsilon_q) ].
@ -253,21 +253,21 @@ which was first introduced by Faleev and co-workers \cite{Faleev_2004,vanSchilfg
The corresponding matrix elements are The corresponding matrix elements are
\begin{equation} \begin{equation}
\label{eq:sym_qsGW} \label{eq:sym_qsGW}
\Sigma_{pq}^{\qsGW}(\eta) = \frac{1}{2} \sum_{r\nu} \qty( \frac{\Delta_{pr\nu}}{\Delta_{pr\nu}^2 + \eta^2} +\frac{\Delta_{qr\nu}}{\Delta_{qr\nu}^2 + \eta^2} ) W_ {p,r\nu} W_{q,r\nu}. \Sigma_{pq}^{\qsGW} = \frac{1}{2} \sum_{r\nu} \qty( \frac{\Delta_{pr\nu}}{\Delta_{pr\nu}^2 + \eta^2} +\frac{\Delta_{qr\nu}}{\Delta_{qr\nu}^2 + \eta^2} ) W_ {p,r\nu} W_{q,r\nu}.
\end{equation} \end{equation}
with $\Delta_{pr\nu} = \epsilon_p - \epsilon_r - \sgn(\epsilon_r-\epsilon_F)\Omega_\nu$ (where $\epsilon_F$ is the energy of the Fermi level). with $\Delta_{pr\nu} = \epsilon_p - \epsilon_r - \sgn(\epsilon_r-\epsilon_F)\Omega_\nu$ (where $\epsilon_F$ is the energy of the Fermi level).
One of the main results of the present manuscript is the derivation, from first principles, of an alternative static Hermitian form for the $GW$ self-energy. One of the main results of the present manuscript is the derivation, from first principles, of an alternative static hermitian form for the $GW$ self-energy.
Once again, in cases where multiple solutions have large spectral weights, self-consistency can be difficult to reach at the qs$GW$ level. Once again, in cases where multiple solutions have large spectral weights, self-consistency can be difficult to reach at the qs$GW$ level.
Multiple solutions of Eq.~\eqref{eq:G0W0} arise due to the $\omega$ dependence of the self-energy. Multiple solutions of Eq.~\eqref{eq:G0W0} arise due to the $\omega$ dependence of the self-energy.
Therefore, by suppressing this dependence, the static approximation relies on the fact that there is well-defined quasiparticle solutions. Therefore, by suppressing this dependence, the static approximation relies on the fact that there is well-defined quasiparticle solutions.
If it is not the case, the qs$GW$ self-consistent scheme inevitably oscillates between solutions with large spectral weights. \cite{Forster_2021} If it is not the case, the qs$GW$ self-consistent scheme inevitably oscillates between solutions with large spectral weights. \cite{Forster_2021}
The satellites causing convergence problems are the above-mentioned intruder states. The satellites causing convergence problems are the above-mentioned intruder states. \cite{Monino_2022}
One can deal with them by introducing \textit{ad hoc} regularizers. One can deal with them by introducing \textit{ad hoc} regularizers.
For example, the $\ii\eta$ term in the denominators of Eq.~\eqref{eq:GW_selfenergy}, sometimes referred to as a broadening parameter linked to the width of the quasiparticle peak, is similar to the usual imaginary-shift regularizer employed in various other theories plagued by the intruder-state problem. \cite{Surjan_1996,Forsberg_1997,Monino_2022,Battaglia_2022}. For example, the $\ii\eta$ term in the denominators of Eq.~\eqref{eq:GW_selfenergy}, sometimes referred to as a broadening parameter linked to the width of the quasiparticle peak, is similar to the usual imaginary-shift regularizer employed in various other theories plagued by the intruder-state problem. \cite{Surjan_1996,Forsberg_1997,Monino_2022,Battaglia_2022}.
However, this $\eta$ parameter stems from a regularization of the convolution to obtain $\Sigma$ and should theoretically be set to zero. \cite{Martin_2016} However, this $\eta$ parameter stems from a regularization of the convolution that yields the self-energy and should theoretically be set to zero. \cite{Martin_2016}
Several other regularizers are possible \cite{Stuck_2013,Rostam_2017,Lee_2018a,Evangelista_2014b,Shee_2021} and in particular, it was shown in Ref.~\onlinecite{Monino_2022} that a regularizer inspired by the SRG had some advantages over the imaginary shift. Several other regularizers are possible \cite{Stuck_2013,Rostam_2017,Lee_2018a,Evangelista_2014b,Shee_2021} and, in particular, it was shown in Ref.~\onlinecite{Monino_2022} that a regularizer inspired by the SRG had some advantages over the imaginary shift.
Nonetheless, it would be more rigorous, and more instructive, to obtain this regularizer from first principles by applying the SRG formalism to many-body perturbation theory. Nonetheless, it would be more rigorous, and more instructive, to obtain this regularizer from first principles by applying the SRG formalism to many-body perturbation theory.
This is one of the aims of the present work. This is one of the aims of the present work.
@ -592,7 +592,7 @@ This yields a $s$-dependent static self-energy which matrix elements read
\\ \\
\times \qty[1 - e^{-(\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2) s} ]. \times \qty[1 - e^{-(\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2) s} ].
\end{multline} \end{multline}
Note that the static SRG-qs$GW$ approximation defined in Eq.~\eqref{eq:SRG_qsGW} is naturally Hermitian as opposed to the usual case [see Eq.~\eqref{eq:sym_qsGW}] where it is enforced by brute-force symmetrization. Note that the static SRG-qs$GW$ approximation defined in Eq.~\eqref{eq:SRG_qsGW} is naturally hermitian as opposed to the usual case [see Eq.~\eqref{eq:sym_qsGW}] where it is enforced by brute-force symmetrization.
Another important difference is that the SRG regularizer is energy-dependent while the imaginary shift is the same for every self-energy denominator. Another important difference is that the SRG regularizer is energy-dependent while the imaginary shift is the same for every self-energy denominator.
Yet, these approximations are closely related because, for $\eta=0$ and $s\to\infty$, they share the same diagonal terms. Yet, these approximations are closely related because, for $\eta=0$ and $s\to\infty$, they share the same diagonal terms.